Digital Signal Processing Reference
In-Depth Information
The perfect mathematical and physical representation is the complex
exponential form
of
the FOURIER- series:
∞
¦
jk
ω
t
xt
( )
=
ce
NB: for k -
∞ ≤
k
≤ ∞
is valid
0
k
k
=−∞
whereby for the complex amplitudes or FOURIER- coefficients the following applies:
1
−
jkt
ω
=
³
c
x t
()
e
dt
0
k
T
0
T
0
The following formulaic derivation of the complex amplitude or the FOURIER- coeffi-
cient
c
k
is an important exercise in complex calculus and should be carried out indepen-
dently “using paper and pencil”.
Illustration 270 already demonstrated the term orthogonality. Accordingly the FOURIER-
series can be represented mathematically and signal-technically in an infinite-dimension-
al vector space. The frequencies
kf
0
are
linear- independent
of each other. In this vector
space they are arranged vertically (orthogonally) one upon the other.
If complex signals on an interval are perpendicular (orthogonal) to each other, they fulfil
the constraint of the following scalar product (the result is a real number):
{
b
0 if
if
mk
mk
≠
=
³
∗
Ψ⋅ Ψ
()
t
()
t
dt
=
α
k
m
a
tT
+
tT
+
tT
+
∗
00
00
00
(
)
(
)
jm
ω
t
jk m
−
ω
t
jk
ω
t
jm
ω
t
jk
ω
t
−
jm
ω
t
³
dt
=
³
³
Let be
Ψ=
( )
t
e
e
e
e
⋅
e
t
=
e
t
0
0
0
0
0
0
n
t
t
t
0
0
0
tT
+
tT
+
tT
+
tT
+
00
00
00
00
(
)
jk m
−
ω
t
j
0
ω
t
0
³
³
³
³
Case 1:
mk
=
e
t
=
e
t
=
edt
=
1
tT
=
α
0
0
0
t
t
t
t
0
0
0
0
tT
+
1
1
00
(
)
jn
ω
t +T
³
jnt
ω
jntt+T
ω
jnt
ω
Case 2:
mk
≠
or
mk n
-
=
e
dt
=
e
=
(
e
0
0
0
−
e
)
=
0
0
0
0
0 0
0
t
jn
ω
jn
ω
t
0
0
0
1
1
1
(
)
(
)
jn
ω
t
jn
ω
T
jn
ω
t
jn
ω
t
(
)
jn
2
π
e
e
−=
1
e
e
−=
1
e
1
−=
1
0
00
0
00
00
0
jn
ω
jn
ω
jn
ω
0
0
0
In the case of k = +/- 1, +/-2, +/-3, ... , two vectors, with equal amplitude, rotate in oppo-
site directions around the zero of the GAUSSian plane as shown in Illustration 305. The
vectorial sum of both vectors results in the chronological form of the real sine wave with
the frequency
kf
0
. This is the only model of sinusoidal (harmonic) oscillation which is not
mathematically and physically inconsistent in itself! The complex
exponential form
of the
FOURIER- series is thus:
∞
¦
jk
ω
t
xt
( )
=
ce
From orthogonality it follows
0
k
k
=−∞
t
+
xt
()
t
+
T
§
∞
·
0
0
0
b
¦
∗
(
)
−
jm
ω
t
jk
ω
t
−
jm
ω
t
³
³
³
Ψ⋅ Ψ
()
t
()
t
dt
=
x t
()
e
dt
=
c e
e
dt
=
0
0
0
¨
¸
k
m
k
a
©
¹
k
=−∞
t
t
